#10055785

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#41058	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,9)(3,4,7,10)(5,8,14,16)(11,12,15,13)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(3,7)(4,10)(11,15)(12,13)(21,29)(25,28)(27,30)(31,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,2)(3,10)(4,7)(5,8)(6,9)(11,13)(12,15)(14,16)(17,20,23,18)(19,24,22,26)(21,28,29,25)(27,31,30,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,10)(2,7,9,3)(5,13,14,12)(8,11,16,15)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,22,23,19)(18,24,20,26)(21,27,29,30)(25,32,28,31)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9962981);